Abstract
Given a filter Δ in the poset of compositions of n, we form the filter in the partition lattice. We determine all the reduced homology groups of the order complex of as Sn - 1-modules in terms of the reduced homology groups of the simplicial complex Δ and in terms of Specht modules of border shapes. We also obtain the homotopy type of this order complex. These results generalize work of Calderbank–Hanlon–Robinson and Wachs on the d-divisible partition lattice. Our main theorem applies to a plethora of examples, including filters associated with integer knapsack partitions and filters generated by all partitions having block sizes a or b. We also obtain the reduced homology groups of the filter generated by all partitions having block sizes belonging to the arithmetic progression a, a+ d, … , a+ (a- 1) · d, extending work of Browdy.
| Original language | English |
|---|---|
| Pages (from-to) | 403-439 |
| Number of pages | 37 |
| Journal | Journal of Algebraic Combinatorics |
| Volume | 47 |
| Issue number | 3 |
| DOIs | |
| State | Published - May 1 2018 |
Bibliographical note
Publisher Copyright:© 2017, Springer Science+Business Media, LLC.
Funding
Acknowledgements The authors thank Bert Guillou and Kate Ponto for their homological guidance and expertise. They also thank Sheila Sundaram, Michelle Wachs, and the referee for essential references. The authors thank Margaret Readdy and the referee for their comments on an earlier draft. Both authors were partially supported by National Security Agency Grant H98230-13-1-0280. This work was supported by a grant from the Simons Foundation (#429370, Richard Ehrenborg). The first author wishes to thank the Princeton University Mathematics Department where this work began. The authors thank Bert Guillou and Kate Ponto for their homological guidance and expertise. They also thank Sheila Sundaram, Michelle Wachs, and the referee for essential references. The authors thank Margaret Readdy and the referee for their comments on an earlier draft. Both authors were partially supported by National Security Agency Grant?H98230-13-1-0280. This work was supported by a grant from the Simons Foundation (#429370, Richard Ehrenborg). The first author wishes to thank the Princeton University Mathematics Department where this work began.
| Funders | Funder number |
|---|---|
| Simons Foundation | 429370 |
| Princeton University | |
| National Security Agency | H98230-13-1-0280 |
Keywords
- Homology
- Partition
- Representation theory
- Symmetric group
ASJC Scopus subject areas
- Algebra and Number Theory
- Discrete Mathematics and Combinatorics